Galactic frame dragging

Do galaxies produce measurable/significant frame dragging effects by their rotation?

I would think frame dragging depends on the mass and speed of rotation... Are such effects only felt at the boundary between rotating gravitational fields and flatter space, i.e. the edge of the milky way? Or would frame dragging produce effects throughout the milky way, i.e. in the gaps between spiral arms as well as between stars.

These effects are completely negligible on the galactic scale. Frame dragging is only significant near the event horizon of a black hole. For comparison, if the entire mass of a galaxy were compressed to a point; its event horizon would be about a million times smaller than the radius of the galaxy, suggesting the effects will be completely negligible.

You're right that the degree of the effect depends only on the mass and speed of rotation (generally measured by the angular momentum). What (little) effects there are would be felt both at the edge of the disk, and between the spiral-arms.